188.8.131.52 Partial Redundancy
Instances in which the system is successful if at least one of n
parallel paths is successful has been discussed. In other instances, at least
k out of n elements must be successful. In such cases, the reliability of the
redundant group (each with the same Probability of Success, p) is given by a
series of additive binomial terms in the form of
Two examples of partial redundancy follow.
A receiver has three channels. The receiver will operate if at
least two channels are successful, that is, if k = 2 or k = 3. The probability
of each channel being successful is equal to p; then
= P (2, 3 | p) + P (3,
3 | p)
= 3p2 (1-p)
Use of the binomial formula becomes impractical for hand
calculation in multi-element partial redundant configurations when the values
of n and k become large.4 In these cases, the normal
approximation to the binomial may be used. The approach can be best
illustrated by an example.
A new transmitting array is to be designed using 1000 RF
elements to achieve design goal performance for power output and beam width. A
design margin has been provided, however, to permit a 10% loss of RF elements
before system performance becomes degraded below the acceptable minimum level.
Each element is known to have a failure rate of 1000 x
10-6 failures per hour. The proposed design is illustrated
in Figure 7.5-11, where the total number of elements is n = 1000; the number
of elements required for system success is k = 900; and, the number of element
failures permitted is r = 100. It is desired to compute and plot the
reliability function for the array.
FIGURE 7.5-11: PARTIAL
For each discrete point of time, t, the system reliability
function, Rs(t) is given by the binomial summation as:
This binomial summation can
be approximated by the standard normal distribution function using Table 7.5-3
to compute reliability for the normalized statistic z.
1 - e-lt
|number of failures|
TABLE 7.5-3: RELIABILITY
CALCULATIONS FOR EXAMPLE 2
F(z) = Rs(t)
Note that Rs(t) = F(z)
By observation, it can be reasoned that system MTBF will be
approximately 100 hours, since 100 element failures are permitted and one
element fails each hour of system operation. A preliminary selection of
discrete points at which to compute reliability might then fall in the 80- to
At 80 hours:
At 100 hours:
These points are then used to plot the reliability function for
the array, shown in Figure 7.5-12. Also shown in the figure are curves for
r=0, 50, and 150.
FIGURE 7.5-12: RELIABILITY
FUNCTIONS FOR PARTIAL REDUNDANT
ARRAY OF FIGURE 7.5-11
4 See any
good textbook on probability and statistics.